Finite Group Databasebeta

← #189155

Subgroups — #189155

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic
6#8657C211yesmaximal, characteristic, complemented
7#8655C211yesmaximal, characteristic, complemented
8#8653C211yesmaximal, characteristic, complemented
9#8651C211yesmaximal, characteristic, complemented
10#8649C211yesmaximal, characteristic, complemented
11#8647C211yesmaximal, characteristic, complemented
12#8645C211yesmaximal, characteristic, complemented
13#8643C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic
15#8042C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17#2984—24no—
18#2984C418yescomplemented
19C2×#446—21no—
20C2×#446—21no—
21C2×#446C2211yescharacteristic
22C2×#446C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×#432—21no—
26C2×#432C2211yescharacteristic
27C2×#431—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#419—21no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1854—21no—
37#1854—21no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1824—21no—
43#1805—21no—
44#1805—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1787—21no—
54C4×#103—21no—
55C4×#103—21no—
56#1736—21no—
57#1574—24no—
58C2×#335C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#334C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70C23×Q8D411yescharacteristic
71C22×M16—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—21no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C414yes—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8Q814yes—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×#117—21no—
83C2×#117—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#446—28no—
89#446—28no—
90#446—22no—
91#446—22no—
92#446C2×C414yes—
93#446C2×C414yes—
94C8×Q8—22no—
95C8×Q8—22no—
96C8×Q8C2×C414yes—
97C8×Q8C2×C414yes—
98#432—22no—
99#432—28no—
100#432C2×C414yes—
101#431—28no—
102#431—22no—
103#431—22no—
104#431C2×C414yes—
105#431D414yes—
106C2×#107—21no—
107C2×#107—21no—
108C2×#106—21no—
109C2×#106—21no—
110#423—22no—
111#423—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8D411yescharacteristic
118C2×C4⋊C8D411yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125#419—44no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—21no—
133C2×#103—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8D411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×#97—21no—
139C2×#97—21no—
140C2×#97—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382—21no—
151#382—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#340—21no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—41no—
181C22×Q8—41no—
182C22×Q8C4⋊C414yes—
183C22×Q8C2×D411yescharacteristic
184C22×Q8C2×D411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8C4⋊C414yes—
214C4×Q8C4⋊C414yes—
215C4×Q8C22×C412yes—
216C4×Q8C22×C412yes—
217C4×Q8C2×Q812yes—
218C4×Q8C2×Q812yes—
219#117—42no—
220#117—42no—
221#117—81no—
222#117—81no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C4⋊C414yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C42—24no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—24no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42C22×C412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×Q812yes—
261C2×C42#4111yescharacteristic
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—81no—
265#107—81no—
266#106—81no—
267#106—81no—
268C4⋊C8—24no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271C4⋊C8—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—42no—
283C4⋊C8C22⋊C414yes—
284C4⋊C8C22⋊C414yes—
285C4⋊C8C22×C412yes—
286C4⋊C8C22×C412yes—
287#103—42no—
288#103—42no—
289#103—42no—
290#103—42no—
291#103—42no—
292#103—42no—
293#103—42no—
294#103—42no—
295C22⋊C8—24no—
296C22⋊C8—42no—
297C22⋊C8C22⋊C414yes—
298C22⋊C8C22⋊C414yes—
299#97—24no—
300#97—24no—
301#97—24no—
302C4×C8—24no—
303C4×C8—24no—
304C4×C8—42no—
305C4×C8C22×C412yes—
306C4×C8C22×C412yes—
307C4×C8C2×D412yes—
308C4×C8C2×D412yes—
309#95—41no—
310#95—41no—
311#95—41no—
312#95—41no—
313C24#12111yescharacteristic
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—24no—
319C2×Q8—28no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8C2×C4⋊C412yes—
330C2×Q8C2×C4⋊C412yes—
331C2×Q8AGL(1,ℤ/8)12yes—
332C2×Q8#13712yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4#11712yes—
360C22×C4#12112yes—
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4C22⋊Q812yes—
365C22×C4C22⋊Q812yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369M16—84no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8C2×C22⋊C412yes—
397C2×C8C2×C22⋊C412yes—
398C2×C8C2×C22⋊C412yes—
399C2×C8C2×C22⋊C412yes—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4C2×C4⋊C412yes—
430C4⋊C4C2×C4⋊C412yes—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—24no—
439C42—82no—
440C42—82no—
441C42C2×C22⋊C412yes—
442C42C2×C4⋊C412yes—
443C42#11712yes—
444C42#12112yes—
445C42C22⋊Q812yes—
446C42C22⋊Q812yes—
447C42C23×C411yescharacteristic
448C42C22×D411yescharacteristic
449C42C22×Q811yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23#38012yes—
458C23C2×#12111yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—22no—
464Q8—48no—
465Q8—41no—
466Q8—41no—
467Q8#42814yes—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4#38012yes—
513C2×C4#38012yes—
514C2×C4#38012yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C22×C4⋊C411yescharacteristic
517C2×C4C2×#11711yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
522C2×C4C2×#13711yescharacteristic
523C2×C4#57511yescharacteristic
524C2×C4#57511yescharacteristic
525C2×C4#57711yescharacteristic
526C2×C4#57711yescharacteristic
527C2×C4#57811yescharacteristic
528C2×C4#57811yescharacteristic
529C8—22no—
530C8—22no—
531C8—22no—
532C8—22no—
533C8—42no—
534C8—42no—
535C8—81no—
536C8—81no—
537C8—81no—
538C8—81no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×#38011yescharacteristic
547C22#304311yescharacteristic
548C22#304411yescharacteristic
549C22#305511yescharacteristic
550C22#305611yescharacteristic
551C22#306711yescharacteristic
552C22#306711yescharacteristic
553C4—22no—
554C4—22no—
555C4—24no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—161no—
563C4—161no—
564C4—161no—
565C4—161no—
566C4#289811yescharacteristic
567C4#289811yescharacteristic
568C4C2×#42811yescharacteristic
569C4#291111yescharacteristic
570C4#305511yescharacteristic
571C4#305611yescharacteristic
572C4#308211yescharacteristic
573C4#308211yescharacteristic
574C2—42no—
575C2#1947211yescharacteristic
576C2#1947311yescharacteristic
577C2#1947811yescharacteristic
578C2#1947911yescharacteristic
579C2#2065111yescharacteristic
580C2#2066111yescharacteristic
581C2#2067811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #8657C2semidirect
7. #8655C2semidirect
8. #8653C2semidirect
9. #8651C2semidirect
10. #8649C2semidirect
11. #8647C2semidirect
12. #8645C2semidirect
13. #8643C2semidirect
18. #2984C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#29842
#219842
#207342
#206472
C2×#15742
#86572
#86552
#86532
#86512
#86492
#86472
#86452
#86432
#80462
#80422